Yoshikawa moves on marked graphs via Roseman's theorem

0.9 MB·

by  Oleg Chterental

YOSHIKAWA MOVES ON MARKED GRAPHS VIA
ROSEMAN’S THEOREM
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OLEG CHTERENTAL
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r
a March 7, 2017
M
6 Abstract. Yoshikawa[Yo]conjecturedthatacertainsetofmoves
on marked graph diagrams generates the isotopy relation for sur-

T] face links in R4, and this was proved by Swenton [S] and Kearton
and Kurlin [KK]. In this paper, we find another proof of this fact
G
for the case of 2-links (surface links with spherical components).
.
h The proof involves a construction of marked graphs from branch-
t
a free brokensurface diagrams,anda versionof Roseman’s theorem
m [R] for branch-free broken surface diagrams of 2-links. As an ap-
[ plication, we find that Brunnian 2-links are ribbon.

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v 1. Introduction
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7 For a smooth oriented manifold M we denote by Diff+(M) the group
1 of orientation-preserving diffeomorphisms of M and by Diff+(M) the
7 0
0 path-component of the identity in Diff+(M). We orient R4 in the
1. standard way. A surface link L ⊂ R4 is a smooth submanifold dif-
0 feomorphic to a closed surface (i.e. compact with no boundary). We
7 say two surface links L and L are isotopic if there is f ∈ Diff+(R4)
1 1 2 0
: with f(L ) = L . A 2-link is a surface link where each component is a
v 1 2
i 2-sphere. Let L be the set of surface links, L+ the set of surface links
X
with orientable components, and L the set of 2-links.
0
r
a In Section 2 we review the definitions of generic projections, broken
surface diagrams, Roseman moves, and Roseman’s theorem. We prove
Theorem 2.9, a version of Roseman’s theorem for branch-free broken
surfacediagramsof2-links, statingthattwoisotopicbranch-freebroken
surfacediagramsofa2-linkarerelatedbyafinitesequence ofRo1, Ro2,
Ro5∗, Ro7, Br1 and Br2 moves (see Figures 3 and 7). This theorem
complements theresults ofTakaseandTanaka[TT], who findexamples
of isotopic branch-free broken surface diagrams of a 2-link that are not
related by Ro1, Ro2, Ro5∗ and Ro7 moves alone.
In Section 3 we review marked graphs in R3, marked graph dia-
grams in R2, Yoshikawa moves on marked graph diagrams, ab-surfaces
obtained from marked graphs, and prove some facts about marked
1
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Figure 1. Non-singular, branch, double and triple
points in a generic projection.
Figure 2. Broken surface diagrams of branch, double
and triple points.
graphsand ab-surfaces. In Section 4 we describe a relationship between
branch-free broken surface diagrams and ab-surfaces, and provide an-
other proof that two marked graph diagrams describe isotopic 2-links
if and only if they are related by a finite sequence of Yoshikawa moves
(see Theorem 4.4). In Section 5 we prove Theorem 5.2 showing that
Brunnian 2-links are ribbon.
Acknowledgements
The author wishes to thank Dror Bar-Natan, Micha l Jab lonowski,
Seiichi Kamada and Vassily O. Manturov for pointing out some inac-
curacies and providing helpful references and comments.
2. Broken surface diagrams and Roseman’s theorem
Definition 2.1 reviews generic projections, broken surface diagrams
and Roseman moves. Roseman’s theorem is stated in Theorem 2.2.
We use Lemma 2.7 to prove Theorem 2.9, a branch-free version of
Roseman’s theorem.
Definition 2.1. References for the definitions we present here can be
found for example in Carter, Kamada and Saito [CKS], Kamada [Kam]
and Roseman [R]. Let π : R4 → R3 be the projection (x,y,z,u) 7→
YOSHIKAWA MOVES ON MARKED GRAPHS VIA ROSEMAN’S THEOREM 3
(x,y,z). If L ∈ L there is L ∈ L isotopic to L such that a neigh-
1 2 1
bourhood in R3 of any point of π(L ) is one of the four possibilities in
2
Figure 1. Such a projection π(L ) will be called generic.
2
Assume L ∈ L is such that D = π(L) is generic. Let sing(D) ⊂ R3
be the set of branch, double and triple points. If p ∈ sing(D) is a dou-
ble point then there exist x 6= x ∈ L with π(x ) = π(x ) and these
1 2 1 2
two points are ordered by the u-coordinates of x and x . Similarly
1 2
if p ∈ sing(D) is a triple point then there exist x 6= x 6= x ∈ L
1 2 3
with π(x ) = π(x ) = π(x ) and these three points are ordered by the
1 2 3
u-coordinates of x , x and x . The orderings extend by continuity to
1 2 3
neighbourhoods of the points in S. The projection D along with all of
the above crossing information is called a broken surface diagram.
Following Takase and Tanaka [TT], we refer to a broken surface dia-
gram with no branch points as branch-free. We consider two broken
surface diagrams equivalent if they differ by the action of Diff+(R3)
0
and agree on crossing information. We may indicate the crossing in-
formation near a double-point by removing a neighbourhood of one of
the pre-images with the convention that missing segments have greater
u-coordinates as in Figure 2. Figure 3 depicts the Roseman moves
on broken surface diagrams. Appropriate crossing information should
be added for completeness. The move Ro5∗ is slightly different but
equivalent modulo the Ro1 and Ro2 moves, to the usual Ro5 move.
If X is a set of surface links let B(X) be the set of broken surface
diagrams corresponding to links in X with generic projections, and let
Bb(X) ⊂ B(X) be the subset of branch-free broken surface diagrams.
△
Theorem 2.2 (Roseman [R]). If D,D′ ∈ B(L) represent isotopic sur-
face links then D and D′ are related by a finite sequence of the moves
in Figure 3 and equivalences in R3.
Remark 2.3. In the paper of Bar-Natan, Fulman and Kauffman [BKF]
there is a proof of the well-known fact that all spanning-surfaces of
a classical link in R3 are tube-equivalent. In that proof, link projec-
tions are used to construct Seifert surfaces via Seifert’s algorithm, and
the main result is deduced by observing how the constructed Seifert
surfaces change when Reidemeister moves are performed on the link
projection.
We wish to approach the proofs of Theorems 2.9 and 4.4 in a similar
manner. We will assign various structures (a set of branch-free broken
surface diagrams in the case of Theorem 2.9 and an ab-surface in the
case of Theorem 4.4) to a broken surface diagram and observe how
4 OLEG CHTERENTAL
Figure 3. Roseman moves on broken surface diagrams,
with crossing information suppressed.
YOSHIKAWA MOVES ON MARKED GRAPHS VIA ROSEMAN’S THEOREM 5
Figure 4. The set π−1(sing(π(L))) ⊂ L in the neigh-
bourhood of a branch point x.
Figure 5. Converting an invalid path to a valid path.
See Definition 2.4 and Remark 2.5.
these structures change when Roseman moves are performed on the
broken surface diagram.
Definition 2.4. Following Carter and Saito [CS], we define a function
cancel : B(L+) → P(Bb(L+)) (where P(X) is the power set of X).
Let D = π(L) be a broken surface diagram corresponding to a generic
projection for some L ∈ L+ with components L = L ∪ ... ∪ L . If
1 r
x ∈ L is such that π(x) is a branch point, then a neighbourhood of
x ∈ π−1(sing(D)) ⊂ L looks as in Figure 4. Let D be the broken
i
surface diagram obtained from D by removing all components except
L . Each D has an equal number of positive and negative branch
i i
points, and any pair of positive and negative branch points can be
−−→
cancelled using an appropriate sequence of Ro6 moves followed by an
−−→
Ro4 move.
Specifically, assume D has 2b ≥ 0 branch points and for 1 ≤ i ≤ r
i i
and 1 ≤ j ≤ b let p ,q ⊂ L be the distinct points such that
i i,j i,j i
π(p ) and π(q ) are positive and negative branch points in D. A pair
i,j i,j
(T = {τ } ,V = v ) is valid if:
i 1≤i≤r S1≤i≤r,1≤j≤bi i,j
- each τ is a permutation of {1,2,...,b },
i i
- V ⊂ L is a disjoint union of embedded oriented compact inter-
i
vals v with endpoints ∂v = {p ,q } and the orientation
i,j i,j i,j i,τi(j)
going from p to q ,
i,j i,τi(j)
- V ∩π−1(sing(D)) ⊂ L is transverse in L and π(V) is embedded
in R3,
−−→
- after performing the sequence of Ro6 moves pushing p along
i,j
v through all intersections in v ∩π−1(sing(D)), it is possible
i,j i,j
−−→
to perform a final Ro4 move cancelling p and q .
i,j i,τi(j)
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The set cancel(D) contains a broken surface diagram E(T,V) for each
valid pair (T,V), obtained by actually performing on D the aformen-
−−→ −−→
tioned Ro6 moves along each v ending in an Ro4 move cancelling the
i,j
branch points p and q . If there are no branch points in D we let
i,j i,τi(j)
cancel(D) = {D}. △
Remark 2.5. It is important to note that the final requirement for valid
pathsinDefinition2.4isnotsuperfluous. Thereexistinvalidcollections
of simple paths satisfying the first three requirements. However if a
particular path satisfies the first three requirements but not the fourth,
then it can be made valid by the modification in Figure 5. This is
depicted on the level of broken surface diagrams in Carter and Saito
[CS, Figure K and Figure M]. △
Remark 2.6. Figure 6 describes P moves on the valid pairs (T,V) of
Definition2.4. Foreachmove, thediagramD = π(L)remainsfixedand
the set π−1(sing(D)) ⊂ L thus remains fixed as well. The moves P2−
P6 correspond to an isotopy of V rel ∂V in L, interacting with the set
π−1(sing(D)). The moves P1andP7describe two types ofinteractions
oftwosimplepathsinV. TheP1moveactsasatranspositionononeof
the permutations in T. These moves are to bethought of only formally,
for the moment. In Lemma 2.7 we will see that the moves P1 − P6
correspond naturally to moves Br1−Br6 on broken surface diagrams,
described in Figure 7. Note that each Br move does in fact represent
an isotopy in R4, since it can be written in terms of Roseman moves,
in particular Ro4 and Ro6 moves. △
Lemma 2.7. a) If D ∈ B(L ) then all valid pairs (T,V) as in Defini-
0
tion 2.4 are related by the P moves in Figure 6.
b) The Br3, Br4, Br5 and Br6 moves can be expressed in terms of
Ro1, Ro2, Ro5∗, Ro7 and Br1 moves.
c) If D ∈ B(L ) and E ,E ∈ cancel(D) then E and E are related by
0 1 2 1 2
the Ro1, Ro2, Ro5∗, Ro7, Br1 and Br2 moves in Figures 3 and 7.
d) If D ,D ∈ B(L ), E ∈ cancel(D ), E ∈ cancel(D ) and D and
1 2 0 1 1 2 2 1
D are related by a Roseman move then E and E are related by a
2 1 2
sequence of Ro1, Ro2, Ro5∗, Ro7, Br1 and Br2 moves.
Proof. a) Assume we have two pairs (T,V) and (T′,V′) with the same
setofpermutationsT = T′ = {τ } andletV = {v }
i 1≤i≤r i,j 1≤i≤r,1≤j≤bi
and V′ = {v′ } . If for some i,j the pairs v and v′ do
i,j 1≤i≤r,1≤j≤bi i,j i,j
not agree in small enough neighbourhoods of the points p and
i,j
q in L, then they can be made to agree using a P6 move, due
i,τi(j)
to the fact that the pairs (T,V) and (T,V′) satisfy the last property
YOSHIKAWA MOVES ON MARKED GRAPHS VIA ROSEMAN’S THEOREM 7
Figure 6. Moves on valid pairs (T,V) for some fixed
broken surface diagram D. See Remark 2.6.
in Definition 2.4. Thus we assume for each i,j the paths v and
i,j
v′ agree in small enough neighbourhoods of the points p and
i,j i,j
q in L. Since each component of L is a sphere, there is an
i,τi(j)
isotopy in L taking v to v′ relative to their common boundary
i,j i,j
p and q . During this isotopy, v does not change in a small
i,j i,τi(j) i,j
enough neighbourhood of its boundary. Such an isotopy can be
accomplished using the P2, P3, P4, P5, and P7 moves. Thus after
performing each isotopy for all choices of i,j we get a sequence of
P moves taking V to V′.
Consider now the case of two pairs (T,V) and (T′,V′) with T 6=
T′. It is enough to assume that all permutations in T agree with
their respective permutations in T′ except for some two permuta-
tions that differ by a transposition. We can induce an arbitrary
transposition using a P1 move. If we wish to perform a P1 move
8 OLEG CHTERENTAL
Figure 7. Br moves for broken surface diagrams, with
crossing information suppressed.
between v and v we can use P4 moves to bring them into the
i,j i,k
exact forminthe left-handside of theP1move. Now if theP1move
is valid (i.e. lifts to the Br1 move in Figure 7), we can perform the
P1 move to induce a transposition. If the P1 move is not valid, it
YOSHIKAWA MOVES ON MARKED GRAPHS VIA ROSEMAN’S THEOREM 9
Figure 8. Showing that a Br6 move can be realized
with Ro1, Br1 and Br4 moves.
can be made valid after a P6 move along either of the paths. Thus
we may induce arbitrary transpositions in the τ ’s using P moves.
i
Thus the P moves suffice to connect all valid pairs (T,V) in Def-
inition 2.4.
b) We leave it to the reader to verify that the Br3, Br4, and Br5
moves can be expressed in terms of Ro1, Ro2, Ro5∗ and Ro7 moves.
Figure 8 expresses a Br6 move using Ro1, Br1 and Br4 moves.
c) If E ,E ∈ cancel(D) then by Lemma 2.7a their defining valid pairs
1 2
are related by P moves. The moves P1, P2, P3, P4 and P5 on
valid pairs can be realized directly by the moves Br1, Br2, Br3,
Br4 and Br5 respectively on broken surface diagrams. The move
P6canberealizedbyasequence involving Ro1, Ro2, Ro5∗, Ro7and
Br6 moves. The P7 move can be realized by a sequence involving
Ro1, Ro2, Ro5∗ and Ro7 moves. By Lemma 2.7b, the Br3, Br4,
Br5 and Br6 moves can be expressed in terms of Ro1, Ro2, Ro5∗,
Ro7 and Br1 moves, so we are done.
d) If D and D are related by an Ro1, Ro2, Ro5∗, or Ro7 move, then
1 2
it is not difficult to see that there are diagrams F ∈ cancel(D ) and
1 1
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F ∈ cancel(D ) such that F and F are related by an Ro1, Ro2,
2 2 1 2
Ro5∗, or Ro7 move. By Lemma 2.7c, E is related to F and E is
1 1 2
related to F by a sequence of Ro1, Ro2, Ro5∗, Ro7, Br1 and Br2
2
moves. Thus there is a sequence of Ro1, Ro2, Ro5∗, Ro7, Br1 and
Br2 moves relating E to E .
−−→ 1 2
If an Ro3 move takes D to D then there are diagrams F ∈
1 2 1
←−−
cancel(D ) and F ∈ cancel(D ) with a Ro1 move taking F to F
1 2 2 1 2
−−→
(note that the two branch points created by the Ro3 move can be
←−−
paired in two obvious ways. One way can be cancelled by an Ro1
←−−
move and the other can be cancelled by Ro1, Br1 and Br4 moves,
similar to Figure 8). Thus by Lemma 2.7c again there is a sequence
of Ro1, Ro2, Ro5∗, Ro7, Br1 and Br2 moves relating E to E .
1 2
IfD andD arerelatedbyanRo4oranRo6move, thencancel(D ) ⊂
1 2 1
cancel(D ) or cancel(D ) ⊂ cancel(D ) and we rely on Lemma 2.7c
2 2 1
once more.
(cid:3)
Remark 2.8. It should be noted that the proof of Lemma 2.7a relies
on the fact that in a sphere, any two simple paths with a common
boundary are isotopic relative to their common boundary. △
Now we can prove a branch-free version of Roseman’s theorem.
Theorem 2.9. If D ,D ∈ Bb(L ) are related by a finite sequence of
1 2 0
Roseman moves then they are related by a finite sequence of Ro1, Ro2,
Ro5∗, Ro7, Br1 and Br2 moves.
Proof. Let D = E ,...,E = D be a sequence of broken surface
1 1 n 2
diagrams with E related to E by a Roseman move for 1 ≤ i < n.
i+1 i
Choose F ∈ cancel(E ) for 1 ≤ i ≤ n. By Lemma 2.7d, F is
i i i+1
related to F by a sequence of Ro1, Ro2, Ro5∗, Ro7, Br1 and Br2
i
moves, for 1 ≤ i < n. Since cancel(E ) = cancel(D ) = {D } and
1 1 1
cancel(E ) = cancel(D ) = {D }, we are done. (cid:3)
n 2 2
3. Marked graphs and ab-surfaces
Lomonaco [L] and Yoshikawa [Yo] described another method of rep-
resenting surface links via certain 4-regular rigid vertex graphs in R3.
Definition 3.1 reviews marked graphs and the Yoshikawa moves on
marked graph diagrams. Definition 3.2 concerns ab-surfaces and ab-
moves. Definition 3.3 presents the thicken and cap functions. Lemma
2.7 describes some key properties pertaining to marked graphs, ab-
surfaces and the thicken and cap functions.